You don't need a new form for each person. You need to send users an editing form. Auto-filling in this case is a form of updating an existing record. This basically means the form would be prefilled with the "submitter" name.
Opening a form’s base URL starts a new record only when the form is configured for new-record creation. To open a specific existing record, you need to add the editing formula to your Airtable. It would generate a unique URL for each record and you can then share those URLs with your clients. That would allow them to click on a link to open their form and fill it.
Make sure to watch the instructional video here. It guides you step-by-step on how to set this up.